Can Aniimo grant swimming and flying?

The mechanism is twining. You bring an Aniimo out with you and take on the physical traversal ability it carries, so what you can cross depends entirely on which one is at your side. Each Aniimo has its own distinct ability, and across the roster those cover sprinting, climbing vertical walls, swimming, diving, gliding, flying, and burrowing underground. A creature is not just a fighter you keep in reserve. It is also the reason a piece of the map stops being closed to you.

Deep water is the plainest case. A body of water too deep to wade through is not a boundary once you have a swimmer twined, it is a route. The same logic runs vertically. A climber puts wall faces in play, a glider gets you down and across, and a flier removes the ground from the problem altogether. Burrowing works the other direction, underneath.

Traversal abilities are rated by level rather than being a flat yes or no. Celestice carries climbing and flying at level one, which is the shape to expect from a creature that has both: two options, each at its floor. That matters when you are picking a companion for a stretch of terrain. Having flying at all and having good flying are separate things, and a level one flier is the entry version of the ability. Plan a route around what your current team can actually do, not around what the ability is called.

Not settled: What a level one traversal ability limits in practice, such as how long you can stay airborne or how far a single flight carries you, is not spelled out.